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When Is an Expression Undefined?

Recognize the two things the GED treats as undefined — dividing by zero and taking an even root of a negative number — and find the values of x that cause them.

Practice this skill 25 min When is an expression undefined? Leer en español
Key idea

What you'll learn

Some calculations simply have no answer. On the GED, an expression is undefined (has no real-number value) in two situations:

  1. Division by zero — any fraction whose denominator equals 0, like \tfrac{5}{0} or \tfrac{3}{x - 4} when x = 4.
  2. An even root of a negative number — like \sqrt{-9} or \sqrt{x - 6} when x is less than 6.

You will learn why these have no answer, how to find the values of x that make an expression undefined (including denominators like x^2 - 16), and what that means in a real-life formula.

Why it matters on the GED: questions like "For what value of x is \tfrac{5}{x-3} undefined?" are quick points, often in the no-calculator section. The idea also appears in harder places — a vertical line has an undefined slope, and formulas like "cost per person" break down when the number of people is zero.

Words to know

  • Undefined — has no value as a real number: \tfrac{7}{0} is undefined.
  • Numerator / denominator — top / bottom of a fraction: in \tfrac{x+1}{x-5} the denominator is x - 5.
  • Real number — any number on the number line: -3, 0, \tfrac{1}{2}, \sqrt{2}, \pi are all real.
  • Radicand — the expression under a root sign: in \sqrt{x - 6} the radicand is x - 6.
  • Even root — a root with an even index: square root \sqrt{\phantom{x}} (index 2), fourth root \sqrt[4]{\phantom{x}}.
  • Odd root — a root with an odd index: cube root \sqrt[3]{\phantom{x}} (index 3). Odd roots of negatives are real: \sqrt[3]{-8} = -2.
  • Excluded value — a value of the variable that makes an expression undefined: for \tfrac{1}{x+2}, x = -2 is excluded.
  • Zero product property — if a \cdot b = 0, then a = 0 or b = 0; used to find where a factored denominator is zero: (x-4)(x+4) = 0 gives x = 4 or x = -4.

All Math words in the glossary

1. Why dividing by zero has no answer

Division undoes multiplication. 12 \div 3 = 4 because 4 \times 3 = 12.

Now try 12 \div 0 = \,?. We would need a number that, times 0, gives 12. But anything times 0 is 0, never 12. No number works, so 12 \div 0 is undefined.

Zero on top is fine. 0 \div 12 = 0, because 0 \times 12 = 0. ✓

ExpressionValueWhy
\tfrac{0}{5}00 \times 5 = 0
\tfrac{5}{0}undefinedno number times 0 gives 5
\tfrac{0}{0}undefinedthe GED treats any zero denominator as undefined
\tfrac{-4}{-2}2negatives are fine

The rule: a fraction is undefined exactly when its denominator equals zero. The numerator does not matter.

A real-life picture: sharing 8 slices of pizza among 4 people gives 2 each, but "sharing among 0 people" makes no sense — there is no "slices per person".

Connection to slope: slope = \dfrac{\text{rise}}{\text{run}}. For a vertical line, the run is 0, so the slope is undefined. For example, through (3, 1) and (3, 7): \dfrac{7 - 1}{3 - 3} = \dfrac{6}{0} → undefined.

Coordinate grid with the vertical line x = 3 through the points (3, 1) and (3, 7). The rise is 7 − 1 = 6 and the run is 3 − 3 = 0, so the slope would be 6/0, which is undefined.Vertical line: run = 0, slope undefined0123456012345678xy(3, 1)(3, 7)rise = 6run = 3 − 3 = 0slope = 6/0→ undefined

2. Finding where a fraction with a variable is undefined

To find the value(s) that make a fraction undefined:

  1. Take only the denominator.
  2. Set it equal to 0.
  3. Solve for the variable.

Linear denominators.

  • \dfrac{5}{x - 3}: x - 3 = 0 \Rightarrow x = 3.
  • \dfrac{x+4}{x+9}: x + 9 = 0 \Rightarrow x = -9. (The x + 4 on top is irrelevant.)
  • \dfrac{12}{3x + 18}: 3x + 18 = 0 \Rightarrow 3x = -18 \Rightarrow x = -6.

Squared or factored denominators can give two values — or one, or none.

  • \dfrac{5}{x^2 - 16}: x^2 = 16 \Rightarrow x = 4 or x = -4 (both make x^2 - 16 = 0).
  • \dfrac{1}{x(x - 5)}: x = 0 or x - 5 = 0, so x = 0 or x = 5.
  • \dfrac{x+1}{x^2 - 6x + 9}: factor, x^2 - 6x + 9 = (x - 3)^2, which is 0 only when x = 3.
  • \dfrac{2}{x^2 + 4}: x^2 is never negative, so x^2 + 4 \ge 4 — it is never 0. This expression is defined for every x.
Number line from −6 to 6 drawn in blue, with open red circles at −4 and 4. The expression 5/(x² − 16) is defined everywhere except x = −4 and x = 4, where the denominator x² − 16 equals 0.5/(x² − 16) is undefined where x² − 16 = 0−6−5−4−3−2−10123456x = −4x = 4defined (blue)(−4)² − 16 = 0 and 4² − 16 = 0, so both values are excluded

Check by substituting. Plug your value into the denominator; you should get exactly 0. For x = -4 in x^2 - 16: (-4)^2 - 16 = 16 - 16 = 0 ✓.

3. Even roots of negative numbers

Any real number squared is zero or positive: 3^2 = 9, (-3)^2 = 9, 0^2 = 0. So there is no real number whose square is -9. That makes \sqrt{-9} not a real number — undefined on the GED. The same is true for every even root (4th, 6th, …).

Careful — these are all fine:

  • \sqrt{0} = 0 (zero under the root is allowed).
  • -\sqrt{9} = -3 (the negative is outside the root).
  • \sqrt[3]{-8} = -2 (odd roots of negatives are real, since (-2)^3 = -8).
  • (-3)^2 = 9 (squaring a negative is always allowed).

Finding where a root is undefined: the radicand must be zero or positive. Set up an inequality.

  • \sqrt{x - 6} is real when x - 6 \ge 0, i.e. x \ge 6. It is undefined when x < 6. (At x = 6 it equals \sqrt{0} = 0 — defined.)
  • \sqrt{2x + 8} is undefined when 2x + 8 < 0 \Rightarrow 2x < -8 \Rightarrow x < -4.
  • \sqrt{1 - x} is undefined when 1 - x < 0 \Rightarrow x > 1. (When you solve -x < -1, dividing by -1 flips the sign.)
Number line from 0 to 12. A closed dot at 6 with a thick arrow to the right shows that the square root of x minus 6 is defined for x greater than or equal to 6; it is not a real number for x less than 6.0123456789101112defined: x ≥ 6not real: x < 6√(x − 6)
Number line from −8 to 2. The part left of −4 is red and labelled undefined, x < −4. A closed blue dot at −4 and the blue part to the right are labelled defined, x ≥ −4. At x = −4 the root is √0 = 0; at x = −5 it would be √(−2), which is not real.√(2x + 8) needs 2x + 8 ≥ 0, so x ≥ −4−8−7−6−5−4−3−2−1012undefined: x < −4defined: x ≥ −4At x = −4: √(2(−4) + 8) = √0 = 0 (defined). At x = −5: √(−2) is not real.

Both rules together. In \dfrac{\sqrt{x+5}}{x-1}, you need x + 5 \ge 0 (so x \ge -5) and x - 1 \ne 0 (so x \ne 1). Check both conditions for every value.

Formula

Rules to memorize (NOT on the formula sheet)

\frac{a}{0} \text{ is undefined for every } a \qquad \frac{0}{b} = 0 \text{ when } b \ne 0 \frac{N}{D} \text{ is undefined} \iff D = 0 \sqrt{r} \text{ (and any even root) is a real number} \iff r \ge 0 \sqrt[3]{r} \text{ (and any odd root) is real for every } r

Here N is a numerator, D a denominator and r a radicand.

None of these rules are on the GED formula sheet. The slope formula m = \dfrac{y_2 - y_1}{x_2 - x_1} is on the sheet; it is undefined when x_2 = x_1 (a vertical line).

Worked example

Example 1 (easy, word problem): zero on top or bottom?

A family has 8 slices of pizza. Which calculation has no answer: "8 slices shared among 0 people" (8 \div 0) or "0 slices shared among 8 people" (0 \div 8)?

Two panels. Left: 0 slices shared by 8 people, shown as 8 empty plates each marked 0, so 0 ÷ 8 = 0 because 0 × 8 = 0. Right: a pizza cut into 8 slices shared by 0 people; slices per person has no answer because no number times 0 gives 8, so 8 ÷ 0 is undefined.0 ÷ 8 = 0, but 8 ÷ 0 is undefined0 slices shared by 8 people00000000each person gets 00 ÷ 8 = 0 because 0 × 8 = 08 slices shared by 0 peopleslices per person = ?no number n has n × 0 = 8
  1. 0 \div 8 = 0, because 0 \times 8 = 0. Each person gets 0 slices — a real answer.
  2. 8 \div 0: we need a number n with n \times 0 = 8. Every number times 0 is 0, so no n works.

Answer: 8 \div 0 is undefined; 0 \div 8 = 0.

Check: The calculator shows a DIVIDE BY 0 error for 8 \div 0 and 0 for 0 \div 8 ✓.

Worked example

Example 2: a linear denominator

For what value of x is \dfrac{x + 2}{4x - 12} undefined?

Table of values for (x + 2)/(4x − 12). x = −2: denominator −20, value 0. x = 2: denominator −4, value −1. x = 3: denominator 0, value undefined (highlighted red). x = 4: denominator 4, value 1.5. Only x = 3 makes the expression undefined.Test values in (x + 2)/(4x − 12)xdenominator 4x − 12value−2−200/(−20) = 02−44/(−4) = −130undefined446/4 = 1.5Only x = 3 makes the denominator 0.x = −2 makes the numerator 0: the value is 0 (defined).
  1. Set the denominator equal to 0: 4x - 12 = 0.
  2. Add 12: 4x = 12.
  3. Divide by 4: x = 3.

Answer: x = 3

Check: 4(3) - 12 = 12 - 12 = 0 ✓. (Setting the numerator to zero, x = -2, gives \tfrac{0}{-20} = 0, which is defined.)

Worked example

Example 3 (work backwards): find the missing constant

The expression \dfrac{5}{2x + k} is undefined when x = -3. What is the value of k?

Graph of the denominator y = 2x + 6 for x from −5 to 1. The line crosses the x-axis at (−3, 0), marked in red: with k = 6 the denominator 2(−3) + 6 equals 0, so 5/(2x + 6) is undefined at x = −3.Denominator 2x + k with k = 6−5−4−3−2−11−4−22468xy = 2x + 6(−3, 0)denominator = 02(−3) + 6 = 0
  1. "Undefined at x = -3" means the denominator is 0 there: 2(-3) + k = 0.
  2. Simplify: -6 + k = 0.
  3. Solve: k = 6.

Answer: k = 6

Check: With k = 6 the denominator is 2x + 6, and at x = -3: 2(-3) + 6 = 0 ✓.

Worked example

Example 4 (word problem): when a formula breaks down

Water flows into a 300-gallon tank at r gallons per minute, while a leak drains 20 gallons per minute. The time to fill the tank, in minutes, is t = \dfrac{300}{r - 20}. For what flow rate is the formula undefined, and what does that mean?

Graph of fill time t = 300/(r − 20) against flow rate r from 0 to 80 gallons per minute. The curve passes through (30, 30) and (50, 10) and shoots upward as r gets close to 20. A dashed red vertical line at r = 20 marks where the formula is undefined: the tank never fills.t = 300 / (r − 20)010203040506070800102030405060t (min)r = flow rate (gallons per minute)r = 20:undefined(30, 30)(50, 10)
  1. Set the denominator to 0: r - 20 = 0, so r = 20.
  2. Meaning: water flows in exactly as fast as it leaks out, so the water level never rises — there is no fill time.

Answer: r = 20 gallons per minute (the tank never fills).

Check: At r = 20: t = \dfrac{300}{20 - 20} = \dfrac{300}{0} → undefined ✓. At r = 50: t = \tfrac{300}{30} = 10 minutes, a sensible answer.

Worked example

Example 5 (GED-hard): two conditions at once

Which of these values make \dfrac{\sqrt{x + 5}}{x - 1} undefined: x = -6, x = -5, x = 1, x = 4?

Number line from −7 to 5. Left of −5 is red (x + 5 is negative under the root). From −5 (closed dot) to the right is blue, except an open red circle at 1 (zero denominator). Labels: x = −6 undefined (√(−1) is not real), x = −5 defined (value 0), x = 1 undefined (divide by 0), x = 4 defined (value 1).√(x + 5) / (x − 1): need x ≥ −5 and x ≠ 1−7−6−5−4−3−2−1012345x = −6: undefined√(−1) not realx = −5: definedvalue 0x = 1: undefineddivide by 0x = 4: definedvalue 1Red: undefined (x < −5 or x = 1). Blue: defined.
  1. Root condition: need x + 5 \ge 0, so x \ge -5. x = -6 fails: \sqrt{-1} is not real.
  2. Denominator condition: need x - 1 \ne 0, so x \ne 1. x = 1 fails: denominator = 0.
  3. x = -5: \dfrac{\sqrt{0}}{-6} = \dfrac{0}{-6} = 0 — defined.
  4. x = 4: \dfrac{\sqrt{9}}{3} = \dfrac{3}{3} = 1 — defined.

Answer: x = -6 and x = 1 make it undefined.

Check: Substitute each value; only -6 (negative under the root) and 1 (zero denominator) break a rule ✓.

Calculator tip

Using the TI-30XS MultiView

  • The TI-30XS gives an error message for both undefined cases: typing 5 ÷ 0 enter shows a divide by 0 error, and 2nd x² (√) (−) 9 enter shows a domain error (no real answer). An error screen is a strong hint the expression is undefined. Press clear to get back.
  • To test a value of x in a denominator, type the denominator with that value, e.g. ( (−) 4 ) x² − 16 enter → 0.
  • Cube roots of negatives work normally: 3 2nd ^ (−) 8 enter → -2.

When NOT to use it: these questions are almost always calculator-free and need just one line of algebra: set the denominator to 0, or set the radicand less than 0. Doing it by hand is faster than testing every choice on the calculator.

Common mistake

Common mistakes

  • Looking at the numerator. Wrong: \dfrac{x - 2}{x + 7} is undefined at x = 2. Right: only the denominator matters; x + 7 = 0 \Rightarrow x = -7. At x = 2 the value is 0, which is defined.
  • Sign slip when solving. Wrong: x + 7 = 0 \Rightarrow x = 7. Right: x = -7. Always substitute back.
  • Thinking \tfrac{0}{n} or \sqrt{0} is undefined. Wrong: "a zero anywhere is a problem". Right: \tfrac{0}{5} = 0 and \sqrt{0} = 0 are both fine; only a zero denominator is undefined.
  • Treating cube roots like square roots. Wrong: \sqrt[3]{-8} is undefined. Right: \sqrt[3]{-8} = -2; only even roots of negatives are not real.
  • Missing the second value. Wrong: \dfrac{5}{x^2 - 16} is undefined only at x = 4. Right: also at x = -4, since (-4)^2 = 16.
  • Flipping the inequality the wrong way. Wrong: \sqrt{x - 6} is undefined for x > 6. Right: for x < 6, where x - 6 is negative. Test a value like x = 0: \sqrt{-6} is not real ✓.
Test strategy

GED test strategy

  • Cover the numerator. Put your finger over the top of the fraction; the answer is hiding in the bottom.
  • Plug in the answer choices. Substitute each choice into the denominator (or radicand). The one that gives exactly 0 (or a negative under an even root) is the answer. This also catches sign errors.
  • "Select all" questions: check every expression separately, and remember the four safe cases: \tfrac{0}{n}, \sqrt{0}, odd roots of negatives, and negatives squared.
  • Squares in the denominator usually mean two answers (x = \pm something) unless it is a perfect square like (x - 3)^2, or a sum like x^2 + 4 that is never zero.
  • In word problems, ask what the zero denominator means in real life (no people, no time, no net flow). The answer choice that describes that situation is usually right.
  • Budget about 45–60 seconds per item; these should be quick.

Check yourself

When is an expression undefined?No calculator

Which expression is undefined?

Answer choices
Hint

Only one kind of zero causes a problem: a zero in the denominator.

Check yourself

When is an expression undefined?No calculatorFill in

For what value of x is the expression below undefined?

\frac{6}{2x - 10}

x =

Type your answer in the box. Fractions like 3/4 are fine.

Hint

A fraction is undefined when its denominator is 0. Set 2x - 10 equal to 0 and solve.

Summary

Recap

  • An expression is undefined when it divides by zero or takes an even root of a negative number.
  • For a fraction, set only the denominator equal to 0 and solve. Squared denominators can give two values (x^2 = 16 \Rightarrow x = \pm 4).
  • For \sqrt{\text{radicand}}, it is undefined when the radicand < 0; \sqrt{0} = 0 is fine.
  • \tfrac{0}{n} = 0, -\sqrt{9} = -3, \sqrt[3]{-8} = -2 and (-3)^2 = 9 are all defined.
  • Vertical lines have undefined slope (run = 0); formulas like "total ÷ people" break down at 0.
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